# Sum of finite Cosine Series

*In the previous topic we discussed about the method of getting the sum of finite sine series. Now we will extend our learning to cosines. In this topic I will give a three simple and precise method how to deal with this kind of problem.*

* Formula:*

Process:

- Express the series in terms of summation expression
- Determine the value of the following;

- n – number of terms which can be determined by using the sum of arithmetic sequence
- x – this is the multiplier

- Use the formula provided and simplify the expression by applying the sum or difference of angle formula.

Sample Problem 1:

*Express the sum of cos1° + cos2° + . . . + cos90° and give the sum as a single trigonometric expression.*

* Solution:*

Following the process mentioned above

a. We can express the expression this way,

b. Clearly, n=90 because there are 90 terms. Also it is obvious that x=1.

c. Using the formula:

and

Using addition formula we have,

Going back to our formula,

*Sample Problem 2:*

*cos2+cos4+ cos6 + . . . +cos60 can be expressed as . Where x,y,z are integers. What is the value of A+B+C?*

To solve this we still use the same process

The equivalent summation expression for this is

n=30 because there are only 30 terms in the series. x=2.

By substitution:

sin30=1/2 and cos31=cos(30+1)

By addition formula of cosine:

cos(30+1)=cos30cos1-sin30sin1

cos(30+1)=cos30cos1-sin30sin1

cos(30+1) =

Going back to our formula:

By simplifying,

*Thus, A+B+C=8*

Now your turn

Without using your calculator,find the sum of cos4+cos8+cos12+. . .+cos180

### Dan

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